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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 15 (2014), No 2, pp. 525-535 DOI: 10.18514/MMN.2014.1173
Principal functions of matrix Sturm-Liouville
operators with boundary conditions dependent
on the spectral parameter
Deniz Katar, Murat Olgun, and Cafer Coskun
Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 15 (2014), No. 2, pp. 525–535
PRINCIPAL FUNCTIONS OF MATRIX STURM–LIOUVILLEOPERATORS WITH BOUNDARY CONDITIONS DEPENDENT ON
THE SPECTRAL PARAMETER
DENIZ KATAR, MURAT OLGUN, AND CAFER COSKUN
Received 18 March, 2014
Abstract. Let L denote operator generated in L2.RC;E/ by the differential expression
l.y/D�y00CQ.x/y; x 2 RC WD Œ0;1/;
and the boundary condition .A0CA1�/Y 0 .0;�/�.B0CB1�/Y.0;�/D 0, whereQ is a matrix-valued function and A0; A1; B0; B1 are non-singular matrices, with A0B1�A1B0 ¤ 0: In thispaper, we investigate the principal functions corresponding to the eigenvalues and the spectralsingularities of L:
2010 Mathematics Subject Classification: 34B24; 34L05; 47A10
Keywords: eigenvalues, spectral singularities, spectral analysis, Sturm–Liouville operator, non-selfadjoint matrix operator, principal functions
1. INTRODUCTION
Let us consider the boundary value problem .BVP/
�u00Cq .x/uD �2u ; x 2 RC ; (1.1)
u.0/D 0 ; (1.2)
in L2 .RC/ ; where q is a complex-valued function. The spectral theory of the BVP(1.1)–(1.2) with continuous and point spectrum was investigated by Naimark [20]. Heshowed the existence of the spectral singularities in the continuous spectrum of theBVP .1:1/–.1:2/ : Note that the eigenfunctions and the associated functions (prin-cipal functions) corresponding to the spectral singularities are not the elements ofL2 .RC/. Also, the spectral singularities belong to the continuous spectrum and arethe poles of the resolvent’s kernel, but are not the eigenvalues of the BVP .1:1/–.1:2/.The spectral singularities in the spectral expansion of the BVP .1:1/–.1:2/ in termsof the principal functions have been investigated in [19]. The spectral analysis ofthe quadratic pencil of Schrodinger, Dirac and Klein-Gordon operators with spec-tral singularities were studied in [2–18]. The spectral analysis of the non-selfadjoint
c 2014 Miskolc University Press
526 DENIZ KATAR, MURAT OLGUN, AND CAFER COSKUN
operator, generated in L2 .RC/ by .1:1/ and the boundary condition
y0 .0/
y .0/Dˇ1�Cˇ0
˛1�C˛0;
where ˛i ; ˇi 2 C, i D 0; 1 with ˛0ˇ1 � ˛1ˇ0 ¤ 0 were investigated in detail byBairamov et al. [9]. The all above mentioned papers related with the differentialand difference equations are scalar coefficients. Spectral analysis of the self-adjointdifferential and difference equations with matrix coefficients are studied in [3,10–13].
Let E be an n-dimensional .n <1/ Euclidian space with the norm k:k and letus introduce the Hilbert space L2 .RC;E/ consisting of vector-valued functions withthe values in E. We will consider the BVP
�y00CQ.x/y D �2y ; x 2 RC ; (1.3)
y .0/D 0; (1.4)in L2 .RC;E/ where Q is a non-selfadjoint matrix-valued function .i. e., Q¤Q�/.It is clear that, the BVP .1:3/–.1:4/ is nonselfadjoint. In [14, 21] discrete spectrumof the non-selfadjoint matrix Sturm–Liouville operator and properties of the prin-cipal functions corresponding to the eigenvalues and the spectral singularities wereinvestigated.
Let us consider the BVP in L2.RC;E/
�y00CQ.x/y D �2y; x 2 RC; (1.5)
.A0CA1�/y0 .0;�/� .B0CB1�/y.0;�/D 0; (1.6)
where Q is a non-singular matrix-valued function and A0; A1; B0; B1 are non-selfadjoint matrices such A0B1 �A1B0 ¤ 0. In this paper, we aim to investigatethe properties of the principal functions corresponding to the eigenvalues and thespectral singularities of the BVP .1:5/–.1:6/ :
2. JOST SOLUTION OF .1:5/
We will denote the solution of .1:5/ satisfying the condition
limx!1
y.x;�/e�i�x D I; � 2CC WD f� W � 2C; Im�� 0g (2.1)
by E.x;�/: The solution E.x;�/ is called the Jost solution of .1:5/.Under the condition
1Z0
x kQ.x/kdx <1 (2.2)
the Jost solution has a representation
E.x;�/D ei�xI C
1Zx
K.x; t/ei�tdt; (2.3)
MATRIX STURM–LIOUVILLE OPERATORS 527
for � 2 NCC, where the kernel matrix function K.x; t/ satisfies
K.x; t/D1
2
1ZxC t
2
Q.s/dsC1
2
xC t
2Zx
tCs�xZtCx�s
Q.s/K.s;v/dvds
C1
2
1ZxC t
2
tCs�xZs
Q.s/K.s;v/dvds: (2.4)
Moreover, K.x; t/ is continuously differentiable with respect to its arguments and
kK.x; t/k � c�.xC t
2/; (2.5)
kKx.x; t/k �1
4
Q.xC t2 /
C c�.xC t2 /; (2.6)
kKt .x; t/k �1
4
Q.xC t2 /
C c�.xC t2 /; (2.7)
where �.x/D1Rx
kQ.s/kds and c > 0 is a constant. Therefore, E .x;�/ is analytic
with respect to � in CC WD f� W � 2C; Im� > 0g and continuous on the real axis [1].Let OE˙.x;�/ denote the solutions of (1.5) subject to the conditions
limx!1
OE˙.x;�/e˙i�x D I; limx!1
OE˙x .x;�/e˙i�x
D˙i�I; � 2 NC˙: (2.8)
Then
WhE.x;�/; OE˙.x;�/
iD�2i�I; � 2C˙; (2.9)
W ŒE.x;�/;E.x;��/�D�2i�I; � 2 R; (2.10)
where W Œf1;f2� is the Wronskian of f1 and f2:Let '.x;�/ denote the solution of (1.5) subject to the initial conditions '.0;�/D
A0CA1�; '0.0;�/D B0CB1�: Therefore '.x;�/ is an entire function of �.
Let us define the following functions:
D˙.�/D '.0;�/Ex .0;˙�/�'0.0;�/E.0;˙�/ � 2 NC˙; (2.11)
where NC˙ D f� W � 2C; ˙ Im�� 0g : It is obvious that the functions DC.�/ andD�.�/ are analytic in CC and C�, respectively, and continuous on the real axis. Thefunctions DC and D� are called Jost functions of L:
528 DENIZ KATAR, MURAT OLGUN, AND CAFER COSKUN
3. EIGENVALUES AND SPECTRAL SINGULARITIES OF L
The resolvent of L defined by
R�.L/f D
1Z0
G.x; t I�/g.t/dt; g 2 L2.RC;E/; (3.1)
where
G.x; t I�/D
�GC.x; t I�/; � 2CCG�.x; t I�/; � 2C�;
(3.2)
and
G˙.x; t I�/D
(�E.x;˙�/D�1
˙.�/'T .t;�/; 0� t � x
�'.x;�/�DTC.˙�/
��1ET .t;˙�/; x � t <1:
(3.3)
We will show the set of eigenvalues and the set of spectral singularities of the operatorL by �d and �ss; respectively.
Let us suppose thatH˙.�/D detD˙.�/: (3.4)
From (2.3) and (3.1)–(3.4), we get
�d D f� W � 2CC; HC.�/D 0g[f� W � 2C�; H�.�/D 0g
�ss D˚� W � 2 R�; HC.�/D 0
[˚� W � 2 R�; H�.�/D 0
; (3.5)
where R� D Rnf0g :We see from that, the functions
KC.�/DODC.�/
2i�E.x;�/�
DC.�/
2i�OEC.x;�/; � 2CC; (3.6)
K�.�/DOD�.�/
2i�E.x;��/�
D�.�/
2i�OE�.x;�/; � 2C�; (3.7)
K.�/DDC.�/
2i�E.x;��/�
D�.�/
2i�E.x;�/; � 2 R�; (3.8)
are the solutions of the boundary problem (1.5)–(1.6), where
OD˙.�/D .A0CA1�/ OE˙x .0;�/� .B0CB1�/
OE˙.0;�/: (3.9)
Now let us assume that
Q 2 AC.RC/ ; limx!1
Q.x/D 0 ; supx2Œ0;1/
he"px Q0.x/ i<1; " > 0: (3.10)
Theorem 1. Under the condition (3.10), the operator L has a finite number ofeigenvalues and spectral singularities, and each of them is of finite multiplicity.
MATRIX STURM–LIOUVILLE OPERATORS 529
4. PRINCIPAL FUNCTIONS OF L
Under the condition (3.10), let �1; :::;�j and �jC1; :::;�k denote the zeros HC
in CC and H� in C� (which are the eigenvalues of L) with multiplicities m1;:::;mjandmjC1;:::;mk; respectively. It is obvious that from the definition of the Wronskian�
dn
d�nW�KC.x;�/;E.x;�/
���D�p
D
�dn
d�nDC.�/
��D�p
D 0 (4.1)
for nD 0;1; :::;mp�1; p D 1;2; :::;j; and�dn
d�nW ŒK�.x;�/;E.x;��/�
��D�p
D
�dn
d�nD�.�/
��D�p
D 0 (4.2)
for nD 0;1; :::;mp�1; p D j C1; :::;k:
Theorem 2. The following formulae:�@n
@�nKC.x;�/
��D�p
D
nXmD0
Fm.�p/
�@m
@�mE.x;�/
��D�p
; (4.3)
nD 0;1; :::;mp�1; p D 1;2; :::;j; where
Fm.�p/D
n
m
!�@n�m
@�n�mODC.�/
��D�p
; (4.4)
�@n
@�nK�.x;�/
��D�p
D
nXmD0
Nm.�p/
�@m
@�mE.x;��/
��D�p
; (4.5)
nD 0;1; :::;mp�1; p D j C1; :::;k; where
Nm.�p/D
n
m
!�@n�m
@�n�mOD�.�/
��D�p
(4.6)
hold.
Proof. We will prove only (4.3) using the method of induction, because the caseof (4.5) is similar. Let be nD 0: Since KC.x;�/ and E.x;�/ are linearly dependentfrom (4.1), we get
KC.x;�p/D f0.�p/E.x;�p/; (4.7)
where f0.�p/¤ 0: Let us assume that 1� n0 �mp�2; (4.7) holds; that is,�@n0
@�n0KC.x;�/
��D�p
D
n0XmD0
Fm.�p/
�@m
@�mE.x;�/
��D�p
: (4.8)
530 DENIZ KATAR, MURAT OLGUN, AND CAFER COSKUN
We will prove that (4.3) holds for n0C 1: If Y.x;�/ is a solution of (1.5), then@n
@�nY.x;�/ satisfiesh�d2
dx2CQ.x/��2
i@n
@�nY.x;�/D 2�n @n�1
@�n�1Y.x;�/Cn.n�1/ @
n�2
@�n�2Y.x;�/:
(4.9)Writing (4.9) for KC.x;�/ and E.x;�/, and using (4.8), we find�
�d2
dx2CQ.x/��2
�gn0C1.x;�p/D 0; (4.10)
where
gn0C1.x;�p/Dn@n0C1
@�n0C1KC.x;�/
o�D�p
�
n0C1XmD0
Fm.�p/n@m
@�mE.x;�/
o�D�p
:
(4.11)From (4.1), we have
W�gn0C1.x;�p/;E.x;�p/
�D
�dn0C1
d�n0C1W�KC.x;�/;E.x;�/
���D�p
D 0:
(4.12)Hence there exists a constant f
n0C1.�p/ such that
gn0C1
.x;�p/D fn0C1
.�p/E.x;�p/: (4.13)
This shows that (4.3) holds for nD n0C1: �
Using (4.3) and (4.5), define the functions
Un;p.x/D
�@n
@�nKC.x;�/
��D�p
D
nXmD0
Fm.�p/
�@m
@�mE.x;�/
��D�p
; (4.14)
nD 0;1; :::;mp�1; p D 1;2; :::;j and
Un;p.x/D
�@n
@�nK�.x;�/
��D�p
D
nXmD0
Nm.�p/
�@m
@�mE.x;��/
��D�p
; (4.15)
nD 0;1; :::;mp�1; p D j C1; :::;k:
Then for �D �p; p D 1;2; :::;j;j C1; :::;k;
l.U0;p/D 0;
l.U1;p/C1
1Š
@
@�l.U0;p/D 0; (4.16)
l.Un;p/C1
1Š
@
@�l.Un�1;p/C
1
2Š
@2
@�2l.Un�2;p/D 0;
MATRIX STURM–LIOUVILLE OPERATORS 531
nD 2;3; :::;mp�1;
hold, where l.u/ D �u00CQ.x/u� �2u and @m
@�ml.u/ denote the differential ex-
pressions whose coefficients are the m-th derivatives with respect to � of the corres-ponding coefficients of the differential expression l.u/: (4.16) shows that U0;p is theeigenfunction corresponding to the eigenvalue � D �pI U1;p;U2;p; :::Ump�1;p arethe associated functions of U0;p [16, 17].U0;p;U1;p; :::Ump�1;p; p D 1;2; :::;j;j C 1; :::;k are called the principal func-
tions corresponding to the eigenvalue �D �p; p D 1;2; :::;j;j C1; :::;k of L:
Theorem 3.
Un;p 2 L2.RC;E/; nD 0;1; :::mp�1; p D 1;2; :::;j;j C1; :::;k: (4.17)
Proof. Let be 0� n�mp�1 and 1� p � j: Using (2.5) and (3.10), we obtain that
kK.x; t/k � ce�
qxCt2 : (4.18)
From (2.3), we get �@n
@�nE.x;�/
��D�p
� xne�x Im�p C c
1Zx
tne�
qxCt2 e�t Im�pdt; (4.19)
where c > 0 is a constant. Since Im�p > 0 for the eigenvalues �p; p D 1;2; :::;j; ofL; implies that�
@n
@�nE.x;�/
��D�p
2 L2.RC;E/; nD 0;1; :::mp�1; p D 1;2; :::;j: (4.20)
So we get Un;p 2 L2.RC;E/ from (4.14) and (4.20) Similarly we prove the resultsfor 0� n�mp�1; j C1� p � k: This completes the proof. �
Let �1; :::;�v and �vC1; :::;�l be the zeros ofDC andD� in R� with multiplicit-ies n1; :::;nv and nvC1; :::;nl; respectively. We can show�
@n
@�nK.x;�/
��D�p
D
nXmD0
Cm.�p/
�@m
@�mE.x;�/
��D�p
; (4.21)
nD 0;1; :::;np�1; p D 1;2; :::;v; where
Cm.�p/D�
n
m
!�@n�m
@�n�mD�.�/
��D�p
; (4.22)
�@n
@�nK.x;�/
��D�p
D
nXmD0
Rm.�p/
�@m
@�mE.x;��/
��D�p
;
532 DENIZ KATAR, MURAT OLGUN, AND CAFER COSKUN
nD 0;1; :::;np�1; p D vC1; :::; l;
where
Rm.�p/D
n
m
!�@n�m
@�n�mDC.�/
��D�p
: (4.23)
Now define the generalized eigenfunctions and generalized associated functionscorresponding to the spectral singularities of L by the following :
Vn;p.x/D
�@n
@�nK.x;�/
��D�p
D
nXmD0
Cm.�p/
�@m
@�mE.x;�/
��D�p
; (4.24)
nD 0;1; :::;np�1; p D 1;2; :::;v;
Vn;p.x/D
�@n
@�nK.x;�/
��D�p
D
nXmD0
Rm.�p/
�@m
@�mE.x;��/
��D�p
; (4.25)
nD 0;1; :::;np�1; p D vC1; :::; l:
Then Vn;p; nD 0;1; :::;np�1; pD 1;2; :::;v;vC1; :::; l; also satisfy the equationsanalogous to (4.16).V0;p;V1;p; :::;Vnp�1;p; pD 1;2; :::;v;vC1; :::; l are called the principal functions
corresponding to the spectral singularities �D �p;p D 1;2; :::;v;vC1; :::; l of L:
Theorem 4.
Vn;p … L2.RC;E/; nD 0;1; :::np�1; p D 1;2; :::;v;vC1; :::; l:
Proof. For 0� n� np�1 and 1� p � v using (2.3), we obtain �@n
@�nE.x;�/
��D�p
� .ix/n ei�pxI C
1Zx
.i t/nK.x; t/ei�ptdt
since Im�p D 0; p D 1;2; :::;v; we find that
1Z0
.ix/n ei�pxI 2dx D 1Z0
x2n D1:
So we obtain Vn;p …L2.RC;E/; nD 0;1; :::np�1; p D 1;2; :::;v: Using the similarway, we may also prove the results for 0� n� np�1; vC1� p � l: �
Now define the Hilbert spaces of vector-valued functions with values in E by
Hn WD
8<:f W1Z0
.1Cjxj/2n kf .x/k2dx <1
9=; ; nD 1;2; :::; (4.26)
MATRIX STURM–LIOUVILLE OPERATORS 533
H�n WD
8<:g W1Z0
.1Cjxj/�2n kg.x/k2dx <1
9=; ; nD 1;2; :::; (4.27)
with the norms
kf k2n WD
1Z0
.1Cjxj/2n kf .x/k2dx;
and
kgk2�n WD
1Z0
.1Cjxj/�2n kg.x/k2dx;
respectively. Then
HnC1 ¤Hn ¤ L2.RC;E/¤H�n ¤H�.nC1/; nD 1;2; :::; (4.28)
and H�n is isomorphic to the dual of Hn:
Theorem 5.Vn;p 2H�.nC1/; nD 0;1; :::np�1; p D 1;2; :::;v;vC1; :::; l:
Proof. For 0� n� np�1 and 1� p � v using (2.3) and (4.24), we get1Z0
.1Cjxj/�2.nC1/ Vn;p 2dx
�M
1Z0
.1Cjxj/�2.nC1/
8<: fE.x;�/g
�D�p
2C :::C �@n
@�nE.x;�/
��D�p
29>=>;
where M > 0 is a constant. Using (2.3), we have1Z0
.1Cjxj/�2.nC1/ .ix/n ei�pxI 2dx <1;
and1Z0
.1Cjxj/�2.nC1/
1Zx
.i t/nK.x; t/ei�ptdt
2
dx <1:
Consequently Vn;p 2H�.nC1/ for 0� n� np�1 and 1� p � v: Similarly, we obtainVn;p 2H�.nC1/ for 0� n� np�1 and vC1� p � l . �
Let us choosen0 Dmaxfn1;n2; :::;nv;nvC1; :::;nlg :
By (4.28), we get the following
534 DENIZ KATAR, MURAT OLGUN, AND CAFER COSKUN
Theorem 6.Vn;p 2H�n0 ; nD 0;1; :::np�1;p D 1;2; :::;v;vC1; :::; l:
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MATRIX STURM–LIOUVILLE OPERATORS 535
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Authors’ addresses
Deniz KatarAnkara University, Faculty of Sciences, Department of Mathematics, Ankara, TurkeyE-mail address: [email protected]
Murat OlgunAnkara University, Faculty of Sciences, Department of Mathematics, Ankara, TurkeyE-mail address: [email protected]
Cafer CoskunAnkara University, Faculty of Sciences, Department of Mathematics, Ankara, TurkeyE-mail address: [email protected]
